Структура і спектральні властивості розподілу значень немонотонної функції канторівського типу

Abstract

Let $f$ be a continuous function defined by equality
$$f\left(\sum^{\infty}_{k=1}
\frac{\alpha_k(x)}{5^k}\right)=\delta_{\alpha_1(x)1} +
\sum^{\infty}_{k=2}\left(\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j(x)j}\right),
$$

where $(\varepsilon_n)$ is a sequence of positive real numbers, $0\leq
\varepsilon_n \leq 1$,

$g_{0n}=g_{4n}=\dfrac{2+\varepsilon_n}{4}$, $g_{1n}=g_{3n}=\dfrac{-\varepsilon_n}{4}$, $g_{2n}=0$, $n =1, 2, \ldots$,

$\delta_{0n}=0$, $\delta_{1n}=\dfrac{2+\varepsilon_n}{4}$, $\delta_{2n}=\dfrac{2}{4}=\delta_{3n}$, $\delta_{4n}=\dfrac{2-\varepsilon_n}{4}$,

$\alpha_k(x)$ is a quinary digit of number $x$.

We study Lebesgue structure and spectral properties of distribution of random variable $Y=f(X)$ for a given distribution of random variable $X$ with independent quinary digits.